#### A wire of length $L=2.0 \mathrm{~m}$ and diameter $d=0.5 \mathrm{~mm}$ is suspended vertically from a ceiling. A weight $W=20 \mathrm{~N}$ is attached to the lower end of the wire, causing it to stretch. If the extension of the wire is $e=2.5 \mathrm{~mm}$, calculate the Young's modulus $(Y)$ of the material.Option: 1 $8.1464 \times 10^8 \mathrm{~N} / \mathrm{m}^2$Option: 2 $9.1464 \times 10^{10} \mathrm{~N} / \mathrm{m}^2.$Option: 3 $8.5 \times 10^{10} \mathrm{~N} / \mathrm{m}^2.$Option: 4 $8.1464 \times 10^{10} \mathrm{~N} / \mathrm{m}^2.$

Given:

• Length of wire $L=2.0 \mathrm{~m}$
• Diameter of wire $d=0.5 \mathrm{~mm}$
• Weight attached $W=20 \mathrm{~N}$
• Extension of wire $e=2.5 \mathrm{~mm}$

Step 1: Calculate the cross-sectional area $A$ of the wire:
$\begin{gathered} A=\frac{\pi d^2}{4} \\ A=\frac{\pi \times\left(0.5 \times 10^{-3}\right)^2}{4} \\ A=1.9635 \times 10^{-7} \mathrm{~m}^2 \end{gathered}$

Step 2: Calculate the original volume $V$ of the wire:
$\begin{gathered} V=A \cdot L \\ V=1.9635 \times 10^{-7} \mathrm{~m}^2 \times 2.0 \mathrm{~m} \\ V=3.927 \times 10^{-7} \mathrm{~m}^3 \end{gathered}$

Step 3: Calculate the stress $\sigma$ on the wire due to the weight:
$\begin{gathered} \sigma=\frac{W}{A} \\ \sigma=\frac{20 \mathrm{~N}}{1.9635 \times 10^{-7} \mathrm{~m}^2} \\ \sigma=1.0183 \times 10^8 \mathrm{~N} / \mathrm{m}^2 \end{gathered}$

Step 4: Calculate the strain $\varepsilon$ in the wire:
$\begin{gathered} \varepsilon=\frac{e}{L} \\ \varepsilon=\frac{2.5 \times 10^{-3} \mathrm{~m}}{2.0 \mathrm{~m}} \\ \varepsilon=1.25 \times 10^{-3} \end{gathered}$
Step 5: Using Hooke's Law, $Y=\frac{\sigma}{\varepsilon}$, calculate the Young's modulus $Y$ of the material:
\begin{aligned} & Y=\frac{1.0183 \times 10^8 \mathrm{~N} / \mathrm{m}^2}{1.25 \times 10^{-3}} \\ & Y=8.1464 \times 10^{10} \mathrm{~N} / \mathrm{m}^2 \end{aligned}
The Young's modulus $(Y)$ of the material is $8.1464 \times 10^{10} \mathrm{~N} / \mathrm{m}^2$.
So, option D is correct